Answer:
99.85%
Step-by-step explanation:
To estimate the probability of a meerkat living less than 16.1 years using the empirical rule, we first need to determine how many standard deviations away from the mean 16.1 years is.
Given values:
- Mean (μ) = 10.4 years
- Standard deviation (σ) = 1.9 years
- Target lifespan = 16.1 years
Let n be the number of standard deviations. Therefore:
[tex]\mu + n\sigma = 16.1\\\\\\10.4 + 1.9n = 16.1\\\\\\1.9n = 5.7\\\\\\n = \dfrac{5.7}{1.9}\\\\\\n = 3[/tex]
Therefore, 16.1 years is 3 standard deviations above the mean.
According to the empirical rule, approximately 99.7% of the data falls within 3 standard deviations of the mean. Therefore, to estimate the probability of a meerkat living less than 16.1 years, we need to add the area to the left of the mean (50%) to the area of 3 standard deviations above the mean (half of 99.7%).
[tex]P(X < 16.1)=50\%+\dfrac{99.7\%}{2}\\\\\\P(X < 16.1)=50\%+49.85\%\\\\\\P(X < 16.1)=99.85\%[/tex]
Therefore, the probability of a meerkat living less than 16.1 years is approximately 99.85%.
